Math Matters: The Geometry of Restoration

Image courtesy of Andrew Williams

If you have been following these blogs or checking in on our website or Facebook posts you’ve noticed that these last few years have been filled with a lot of restoration here at Rockport Marine. It occurred to me that there’s a part of that restoration process that is almost completely hidden from view.

Photo by Billy Black

Comprehensive restoration is about returning to a yacht to a prior condition. Yes we often make “improvements” with respect to systems, but a strict constructionist’s reading of the restoration Constitution would demand that we replace the word “improvements” with “compromises,” and for my part I’m inclined to concede the point. You’ll occasionally come across the phrase “better than new restoration..." which is a concept that must have been invented by a broker looking to sell a boat, or an owner trying to explain to his accountant where those particular dollars went exactly. The line gets blurred even further by those of us who have adhered strictly to an original design but executed the work to a level of finish and polish that the original vessel never carried and the original designer probably never intended. I think this is what RMI’s head boatbuilder, John England, is referring to when he talks about “professional arrogance." Still, returning to a prior condition was the goal; we all get a little carried away at times.

What "getting carried away" looks like. Photos by Alison Langley

When it comes to the geometry of a restoration, we’re interested in one prior condition in particular, that is, “original condition.” But even that can be tricky. Do we have the designer’s original lines plan, or better yet, a table of offsets? Even if we do, given the nature of printed and photocopied media, are there concerns about distortion? Was the boat even built to the original lines plan? Has it been altered or modified in some way since? Have the ravages of time and decay altered the shape? How can we tell? There are a lot of questions, and if you’re to have any hope of answering them you have to measure the boat.

It turns out there are several ways to do it. The best way depends on the available time and budget, where the vessel is located, and the degree of accuracy required. Broadly speaking there are three reliable ways to measure a boat; you can do it manually, you can do it with a camera, or you can do it with light. For a variety of reasons we don’t use the first two methods. For that reason and because the manual and photogrammetric methods have already been described well in WoodenBoat Magazine, I’m just going to focus on the last one.

3-D imaging is the process of using light to take accurate measurements. Specifically, it refers to the process of directing laser beams in a controlled manner at an object, then obtaining extremely accurate distance measurements from you to that object from a multitude of angles. This is lots of measurements. Lots, like millions. It can be used to measure a computer chip or an ocean floor. We use it to measure boats, or rather we use Andrew Williams, and he uses it to measure boats for us.

The product or output of this measurement process is a series of points, each with an xyz-coordinate. Remember Cartesian coordinates from high school and college?

Well the box is really big, and the grid is really small, and the file size is huge. Like 60 gigabytes huge. That’s partly because it registers millions of points on the hull, and partly because the equipment doesn’t know you just want the boat so it measures everything. I mean everything. It’s called a point cloud. When Andrew first measured Bolero inside our shop, the whole building was in the point cloud. The bandsaw was there, the table saw was there, the gantry crane was there, you could even see where people had been standing. It reminded me of scenes from The Matrix.

Point cloud image in the rough. Courtesy of Andrew Williams

After measuring Andrew begins what he calls “thinning”, essentially throwing out the millions of points you’re not interested in, and the file size comes down. Next he creates something called a mesh from the points, and again the file size comes down. A mesh is a series of polygons connecting every point in the point cloud. The resulting surface, or collection of polygons, can look crude or quite refined depending on the number of points. Take a look at these polygon meshes of a hand and you’ll see what I mean.

Andrew has to measure a bigger surface to be sure, but also has far less rapid shape change and millions of points to work with. At this stage in the process he is able to do some smoothing or fairing as needed, always with the ability to limit allowable deviation from the measured points. The final step is to convert the mesh to Non-uniform rational B-Spline or NURBS geometry. It’s here that the filesize drops to a useful size of a few megabytes. I have to confess I can’t explain this last process of converting a polygon mesh to a NURBS surface. It requires a level of mathematical understanding that I do not have, but sufficient to say this last part is done with specialized software. NURBS geometry is what’s used by design professionals to design everything from cell phones and cameras, to jewelry and yachts. If you’ve ever seen a computer model rendering of anything that doesn’t yet exist, odds are it’s due to the work of Pierre Bezier and Paul de Casteljau, creators of NURBS modeling program.

In the end what we have is a computer model of the boat. It’s a simple matter to reproduce the designer’s grid of waterlines, sections, and diagonals and project them onto the hull model. Print this at a corresponding scale on transparent media, and you can overlay the existing lines on the original lines to analyze how the hull shape differs from the original boat, or at least from the boat the designer originally intended. Furthermore, if we thought to record where the boat was floating before we took it out of the water we can use the model to rapidly and accurately calculate, moment to trim, sinkage, displacement/weight, wetted surface area, and various performance ratios. Armed with that data we’re much better placed to enter the conversation about whether those proposed changes are improvements or compromises after all.